Figure 1.
On each trial two 50 ms tones, separated by an interval of 950 ms, were played and the participant was asked to respond which of the 2 tones was higher by pressing a button. Immediately after the button press, visual feedback in the form of a smiling face for correct answers, and a sad face for incorrect answers was presented for 300 ms. The inter-trial-interval was 700 ms.; The two frequencies were drawn from a wide distribution and their ratio was determined by a staircase paradigm (see Materials and Methods).
Figure 2.
Performance of participants in Experiment 1.
A. Pattern of responses. Each dot corresponds to one trial of one participant, where the axes denote the frequencies of the 2 tones in the trial: the abscissa is the frequency of the 1st tone, , and the ordinate is the frequency of the 2nd tone,
, both on a logarithmic scale. The color of the dot denotes the outcome of the trial: correct responses are denoted by blue and incorrect responses by red. The vertical and horizontal lines correspond to the lines in which
and
, respectively. The diagonal line corresponds to the line in which
. These lines partition the
space into different regions, denoted using a different background color. The numbers in each region denote the fraction of correct responses in the region ± SEM. Note that the pattern of correct responses is not symmetrical with respect to the diagonal, as expected from a participant whose probability of success in the trial depends solely on the ratio of the two frequencies. B. A two-dimensional histogram of performance rate, computed by binning the data presented in A and computing the fraction of correct responses in each bin. Bins in which the number of trials was smaller than 50 were not analyzed and are colored green. Note in particular the 2 squares marked by arrows. Although they are of equal ‘objective’ difficulty (they are located at the same distance from the diagonal), performance differed substantially: in the square denoted by the upper arrow performance was at chance level (50.8% correct responses) whereas in the square denoted by the lower arrow it was 92.3%.
Figure 3.
To estimate the effect of stimuli administered in previous trials on decision in a trial, we fitted a linear non-linear model that relates the outcome of each trial to a linear combination of present and past stimuli (Eq. 1). The parameters that minimize the square error between the prediction of the model and participants' responses are presented. Green - , Dark blue -
, Black -
. Error bars are 68% confidence intervals (equivalent to one standard deviation in a normal distribution) and we assumed that
, which means that the model had 9 free parameters (
and
), and was fitted using 16,380 trials (65 trials in 252 blocks).
Figure 4.
The parameters of the implicit memory model, the standard deviation of the noise, and the memory weight,
were estimated for each of our experimental blocks to minimize the square error between the model and the observed behavior. These parameters were used to simulate the behavior of an implicit-memory participant in that block. The results of the simulation are presented in A and B, (same presentation as in Fig. 2A and 2B, respectively). Note the similarity between Figs. 4A and 2A and between Figs. 4B and 2B, indicating that the implicit-memory model can account for the contraction bias. C, Estimation of the recency effect in the implicit memory model. Same analysis as in Fig. 3.
Figure 5.
A, Pattern of responses in the Bias+ condition, in which the fraction of trials in the Bias+ region (yellow) is larger than the fraction of trials in the Bias− region (gray). B, Pattern of responses in the Bias− condition, which oversamples the Bias− region. Same presentation as in Fig. 2A. C, Experimental (black) and Implicit Memory Model simulation (purple) Mean ± SEM JND in the Bias+ (left) and Bias− (right) conditions. In the simulations, the parameters of each block were estimated in the Bias+ condition and were used to simulate the implicit memory model in both the Bias+ and in the Bias− conditions.